Desynchronization is a process inverse to synchronization,
where initially synchronized oscillating systems desynchronize
as parameters change or they do so under the influence of an external force
or feedback. Desynchronization is important, for example, in neuroscience
and medicine, where pathologically strong synchronization of neurons
may severely impair brain function as, e.g., in Parkinson's disease or
epilepsy.

This article briefly overviews several methods for the control of
(de)synchronization in oscillatory networks.

The phases \(\psi_{j}\) characterize the oscillatory dynamics of the
elements of the ensemble and increase by \(2\pi\) after each
completed cycle. For oscillating neurons, for instance, the cycle
can be defined as the time period between two
successive spikes or bursts.

For the inhomogeneous ensemble (1), i.e., if
the natural frequencies \(\omega_j\) are different, the phase oscillators
remain desynchronized and oscillate with different frequencies
if the coupling among the oscillators (parameter \(C\)) is sufficiently weak.
The desynchronization-synchronization transition takes place
in system (1) if the coupling among the oscillators
increases. In the limit \(N \to \infty\) and if the natural
frequencies \(\omega_j\) are randomly chosen from a unimodal symmetric
probability density
\(g(\omega)\ ,\) \(g(\Omega+\omega)=g(\Omega-\omega)\ ,\)
where \(\Omega\)
is the mean frequency, the critical coupling of spontaneous
synchronization is given by (Kuramoto 1984, Strogatz 2000)
\[\tag{2}
C_{cr} = \frac{2}{\pi g(\Omega)}.
\]

For \( C < C_{cr} \ ,\) the system relaxes to an incoherent state,
where all oscillators are not synchronized, but for \(C > C_{cr}\ ,\) mutual
synchronization occurs in a group of oscillators.
This transition can be characterized by values of the
order parameter \(R(t)\) calculated as
\[\tag{3}
R(t)\exp(i\Psi(t)) = \displaystyle{\frac{1}{N}}
\sum_{j = 1}^{N}\exp(i \psi_j(t)),
\]

where \(\Psi(t)\) is the mean phase. The state of in-phase synchronization,
were all phases are close to each other \(\psi_{j} \approx \psi_{k}\ ,\) is
characterized by large values of \(R \approx 1\)
( Figure 1). For a desynchronized state,
where the phases are uniformly distributed on the
circle \((0, 2\pi)\ ,\) the order parameter is small,
\(R \approx 0\ ,\) and scales as \(R \sim 1/\sqrt{N}\)
with the number of elements \(N\) --
so-called finite-size effect (Pikovsky et al. 2001).

The problem of desynchronization consists in designing a method which
can effectively desynchronize an ensemble of strongly synchronized
oscillators by delivering a stimulation signal \(S(t)\) to the oscillators.
The efficacy and the main properties of several desynchronization
methods will be illustrated below on exemplary phase ensemble
(1). However, the discussed methods have successfully
been applied to more complicated and realistic models including neuronal
ensembles. Some of them have been tested experimentally as well, see the
bibliography for further reading at the end of the corresponding sections.

Single-pulse stimulation

Figure 2: Two identical single pulses are delivered at the ensemble's vulnerable phase (Tass 2001b). The first single pulse hits the ensemble in a stably synchronized state and causes a desynchronization. The second single pulse is delivered to the ensemble in a weakly synchronized state. Instead of causing a desynchronization, the second single pulse strongly synchronizes the ensemble.

A single pulse of appropriate strength delivered at
a vulnerable phase of a fully synchronized population of phase
oscillators desynchronizes the population both in the absence
(Winfree 1980) and in the presence (Tass 1999)
of noise, see Figure 2,
where the time course of the firing density \(p(t)\ ,\)
the average number density of the oscillators which have zero phase at
time \(t\) is shown. The vulnerable phase denotes the
critical value of the mean phase \(\Psi_{cr}(t)\)
of synchronized ensemble at which the population is most
capable to exhibit stimulation-induced desynchronization
(Tass 1999, see also Winfree 1977).
The strength of an effectively
desynchronizing pulse is typically weaker than that of a resetting
single pulse. Effective desynchronization, however, requires
that the vulnerable phase and the stimulus strength
are thoroughly calibrated. Hence, variations of system parameters
require a recalibration. There is another limitation of this stimulation
technique that is relevant, in particular, for possible clinical
applications to deep brain stimulation (Tass 2000).
The effect of a single pulse crucially depends on the initial
conditions of the population.
For desynchronization a single pulse has to hit the
fully synchronized population at the vulnerable phase
( Figure 2). Delivered to an only partially
or weakly synchronized population, such a pulse will cause
a synchronization ( Figure 2) (Tass 2001b).
Hence, such a stimulus cannot be used to reliably prevent
from resynchronization.

Bibliography for further reading: (Zhai et al. 2005).

Double-pulse stimulation

Figure 3: Two identical double pulse stimuli are delivered to the initially stably synchronized population (Tass 2001b). Both double pulse stimuli desynchronize the population -- irrespective of the population's initial state at stimulus onset. Accordingly, double-pulse stimulation enables to block the resynchronization.

Motivated by possible clinical applications, stimulation techniques
have been developed which robustly cause a desynchronization
-- irrespective of the initial state of the neuronal population
at stimulus onset (Tass 2001b). For this purpose, the
double-pulse method uses two qualitatively different stimuli:
The first, stronger pulse resets the collective dynamics,
irrespective of its initial state at stimulus onset. The second,
weaker pulse follows after a calibrated delay; it hits the population
in the vulnerable phase and causes a desynchronization
(Tass 2001a, Tass 2001b). The pause between the first and the second
pulse as well as the strength of, in particular, the second pulse
have to be calibrated thoroughly. Instead of a strong first pulse
which causes a hard reset, i.e., a rapid reset achieved within at
most a cycle of the synchronized oscillation, alternatively, one
can also utilize a soft reset (Tass 2002a, Tass 2002b).
The latter is typically achieved by a brief low-frequency entraining
pulse train, which causes a reset within several cycles
of the synchronized oscillation. A soft reset may be superior,
e.g., whenever strong stimuli may cause tissue damage
(Tass 2002a, Tass 2002b). However, with double-pulse
stimulation one still has to face the problem that biologically
inevitable variations of system parameters require frequent
recalibration.

Coordinated reset stimulation

Figure 4: Mechanism of action of CR: Brief and mild resetting stimuli are administered at different sites at subsequent times and cause an effective transient desynchronization. Desynchronized firing of neurons is maintained by repetitive administration of CR stimuli.

The motivation behind the development of coordinated reset stimulation
was to design a robustly desynchronizing stimulation technique
which is considerably more robust with respect to variations of system
parameters as opposed to single- and double-pulse stimulation.

The mechanism of action of coordinated reset (CR) stimulation is
schematically shown in Figure 4.
The main idea behind CR stimulation is to achieve
a desynchronization indirectly, by shifting the network
into an unstable state, from where it transiently relaxes into a
desynchronized state. For this, weak phase resetting stimuli are
sequentially delivered at different sites (Tass 2003a, Tass 2003b).
In this way the oscillatory population is split into
distinct clusters related to the different stimulation sites, respectively.
In the case of electrical stimulation of brain tissue the resetting
stimuli are brief high-frequency pulse trains. The latter are
sequentially delivered with a delay of \(T/n\ ,\) where \(n\)
is the number of the stimulation sites, and \(T\) approximates
the mean period of the ensemble. \(T = 2\pi /\Omega\ ,\)
where \(\Omega\) is the mean frequency of
the ensemble.

Modelling the effect of electrical pulse stimulation on the
phase dynamics of a neuron, the stimulation
signal for the phase oscillator \(j\) from the ensemble
(1) attains the form \(S_{j}(t) = I X(t)\cos(\psi_{j})\ ,\)
where \(I\) is the intensity of the stimulation and \(X(t) = 1\)
if the oscillator \(j\) is stimulated at time \(t\)
and \(X(t) = 0\)
otherwise (Tass 1999, Tass 2003b, Tass 2003a).
The term which has to be added to the right-hand side of
Eq. (1) then takes the form
\[\tag{4}
\sum_{m = 1}^{n} I_{m} \rho_{m,j} X_{m}(t) \cos(\psi_{j}),
\]

where \(X_{m}(t)\) is the pulse train administered to the
\(m\)th
stimulation site and parameters \(\rho_{m,j}\) model the
spatial profile of the current spread.

where \(R_n(t)\) and \(\Psi_n(t)\) are the corresponding real amplitude
(order parameter of the degree \(n\)) and real phase,
where \(0 \le R_n(t) \le 1\) for all times \(t\ .\)
Cluster variables are convenient for characterizing synchronized states of
different types: Perfect in-phase synchronization corresponds to
\(R_1 = 1\) (\(R_{1}\) equals the order parameter \(R\) from
Eq. (3)), whereas an incoherent state
with uniformly distributed phases is associated with \(R_n = 0\)
(\(n = 1, 2, 3, \dots\)). Small values of \(R_1\)
combined with large values of \(R_n\) are indicative
of an \(n\)-cluster states consisting
of \(n\) distinct and equally spaced clusters, where
all oscillators have similar phase within each cluster, see Figure 5,
where \(R_{1} \approx 0.02\) and \(R_{4} \approx 0.87\ .\)

The stimulation mechanism of CR stimulation
is as follows ( Figure 4): After a few periods of
stimulation the oscillatory population is shifted into a cluster state.
After switching off the stimulation, the ensemble returns to
a synchronized state, on this way transiently passing through a uniformly
desynchronized state. This procedure is repeated, so that the
ensemble is kept in a desynchronized state. The repetitive
stimulus administration can be performed either regardless of the state
of the stimulated ensemble (open loop control), or in a demand-controlled
way (closed loop control). In the later case, e.g.,
either identical stimuli are delivered whenever \(R_1\) exceeds a critical
threshold ( Figure 6, lower plot), or
CR stimuli are periodically delivered with a stimulus duration
that is adapted to \(R_{1}\) ( Figure 6, upper plot).

Linear multisite delayed feedback

Figure 7: The macroscopic activity (mean field) of the controlled population is measured, delayed, amplified and fed back in a spatially coordinated way via several stimulation sites using different delays for different stimulation sites.

Similarly to coordinated reset stimulation, the linear multisite
delayed feedback is delivered via several stimulation sites,
e.g. via four sites as illustrated in Figure 7, where separate stimulation
signals are applied via each site. These separate stimulation signals
are derived from the delayed mean field of the ensemble by using
different time delays for the different stimulation signals. The mean
field characterizes the collective macroscopic dynamics of the
oscillators and can be viewed as the ensemble average of the signals
of individual oscillators \(Z(t) = N^{-1}\sum_{j=1}^{N} z_j(t)\ ,\)
where \(z_j(t)\ ,\) \(j = 1, \dots , N\) are the signals
of the individual
elements of the ensemble. In the phase representation,
\(Z(t)\) attains the form of \(Z_{1}(t)\)
from Eq. (5).

The delays \(\tau_m\) are symmetrically distributed
with respect to the main delay \(\tau\ ,\) where the smallest time distance
between neighboring stimulation sites is chosen as \(\tau/4\ .\)
In the case \(\tau = T\) (mean period of the ensemble),
the delays \(\tau_m\) are uniformly distributed over the mean period
\(T\ .\)

Figure 8: Upper-left plot: Distribution of the delayed phases \(\Psi(t-\tau_m)\) of the stimulation signals \(S_{m}(t)\) of the multisite delayed feedback for four stimulation sites and for \(\tau = T\) in Eq. (6). The other plots illustrate the stimulation-induced clustered states indicated by the time-averaged order parameters \(\langle R_{1} \rangle\ ,\) \(\langle R_{2} \rangle\ ,\) and \(\langle R_{4} \rangle\) (encoded in color) versus delay \(\tau\) and stimulus amplification \(K\ .\)

In the framework of the phase ensemble (1)
the stimulation signals attain the form
\(S_{m}(t) = K R(t - \tau_{m}) \sin(\Psi(t - \tau_{m}) - \psi_{j}(t))\)
and the term added to the right-hand side of Eq. (1)
reads
\[\tag{7}
\sum_{m = 1}^{n} \rho_{m,j} S_{m}(t),
\]

where the parameters \(\rho_{m,j}\) play the same role as for the
coordinated reset stimulation (see Eq. (4)).
Assuming that the mean phase \(\Psi(t)\) rotates with a constant
frequency \(\Omega\ ,\) the delayed mean phases
\(\Psi(t-\tau_{m})\) are uniformly distributed on the unit circle
(see Figure 8, upper left plot). Then the phases
\(\psi_{j}(t)\) of the sub-population assigned to the stimulation site
\(m\) are attracted to the phase \(\Psi(t-\tau_{m})\) of the corresponding stimulation signals. Hence, the phases of all oscillators stimulated
with the multisite delayed feedback are redistributed symmetrically
on the circle \((0,2\pi)\) in a cluster state. The order parameter
\(R_{1}(t)\) gets thus minimized. Depending on the values of delay
\(\tau\) in Eq. (6), the stimulation can induce phase clusters in the stimulated ensemble, where the corresponding order parameter \(R_{m}\) attains large values, see Figure 8. For example, for
the optimal value of the delay \(\tau = T\) the stimulation
induces a four-cluster state (for four stimulation sites),
where \(R_{1}\) and \(R_{2}\) are small and \(R_{4}\) is larger. For other
values of \(\tau\ ,\) for instance, for \(\tau = 2T\) the stimulated
ensemble exhibits a two-cluster dynamics, where \(R_{1}\) is small
and \(R_{2}\) is large. The cluster states become less pronounced,
and the phases redistribute on the circle even more uniformly
if in the model ensemble a local coupling as well as spatially
decaying profile of the current spread is taken into account
(Hauptmann et al. 2005a).

An important property of the linear multisite delayed feedback
stimulation is its inherent demand-controlled character.
As soon as the desired desynchronized state is achieved,
the values of the order parameter
\(R(t)\) become small and, thus, the amplitude of the stimulation signals
\(S_{m}(t)\) becomes small as well. The stimulated ensemble is then
subjected to a highly effective control at a minimal amount of
stimulation force.

Linear single-site delayed feedback

Figure 9: Impact of the single-site linear delayed feedback on the collective dynamics of the stimulated oscillators (1) versus delay \(\tau\) and stimulus amplification \(K\ .\) The values of the time-averaged order parameter \(\langle R \rangle\) (see Eq. (3)) are encoded in color ranging from red (synchronization) to blue (desynchronization).

Stimulation with linear single-site delayed
feedback utilizes only one recording site and, in contrast to the above
methods, only one stimulation site.
In a first approximation, all oscillators of the ensemble are stimulated
with the same stimulation signal \(S(t)\) constructed from the delayed
mean field of the ensemble \(S(t) = KZ(t-\tau)\ .\) For the coupled
phase oscillators (1) the stimulation signal
\(S(t) = KR(t-\tau)\sin(\Psi(t-\tau) - \psi_{j}(t))\) is added to the
right-hand side of Eq. (1).

Depending of the values of the stimulation parameters \(\tau\)
and \(K\) the
stimulation can result either in an enhancement or in a suppression
of synchronization in the stimulated ensemble. The impact of the
stimulation is illustrated in Figure 9.
Red is the region of parameters \((\tau,K)\ ,\) where the extent
of synchronization is enhanced (large values of \(R\)). Blue
indicates regions of desynchronization (small values of \(R\)).
In the limit \(N \to \infty\) the order parameter \(R = 0\) in the
desynchronization regions, where the phases are uniformly distributed
on the circle \((0,2\pi)\)
(Rosenblum & Pikovsky 2004a, Rosenblum & Pikovsky 2004b).
This is the
state of complete desynchronization, where the stimulated oscillators
rotate with different frequencies indicating an absence of any ordered
state whatsoever. Along with the order parameter, in the desynchronized
state the amplitude of the stimulation signal \(S(t)\) vanishes as well.
In this sense, the stimulation with linear single-site delayed
feedback represents a noninvasive control method for desynchronization
of coupled oscillators.

Nonlinear delayed feedback

Figure 10: The macroscopic activity (mean field) of the controlled population is measured, delayed, nonlinearly combined with the instantaneous mean field, amplified, and fed back via a single stimulation site.

As in the case of the linear single-site delayed feedback, for the
stimulation with nonlinear delayed feedback (NDF) only one registration and
one stimulation site are required, see Figure 10.
All stimulated oscillators receive
the same stimulation signal \(S(t)\) which is constructed from the
mean field of the ensemble. It is assumed that the measured mean
field \(Z(t)\) of the ensemble has the form of a complex analytic
signal \(Z(t) = X(t) + iY(t)\ ,\) where \(X(t)\)
and \(Y(t)\) are the
real and imaginary parts of \(Z(t)\ ,\) respectively. In the case if only
a real part \(X(t)\) of the mean filed is measured, the imaginary part
can be constructed, e.g., with the help of the Hilbert transform.

The stimulation signal is then constructed by a nonlinear combination
of a delayed complex conjugate mean field with the instantaneous mean field
\[\tag{8}
S(t) = K Z^2(t)Z^{\ast}(t - \tau),
\]

where \(K\) is a stimulus amplification parameter,
\(\tau\) is a time delay,
and the asterisk denotes complex conjugacy. For the phase oscillators
(1) the stimulation signal added to the right-hand
side of Eq. (1) attains the form
\[\tag{9}
S(t) = KR^2(t)R(t-\tau) \sin \left ( 2\Psi(t) -
\Psi(t-\tau)-\psi_{j}(t) \right ).
\]

The impact of the NDF on the stimulated
oscillators is twofold. On one hand, the stimulation can effectively
desynchronize even strongly interacting oscillators for a broad range
of values of the stimulus amplification \(K\ ,\) see Figure 11, left plot. This effect is very robust
with respect to the variation of the delay \(\tau\) and, as a result,
with respect to the variation of the mean frequency \(\Omega\) of
the stimulated ensemble. On the other hand,
in a weakly coupled ensemble the stimulation can induce synchronization
in small, island-like parameter regions complemented by wide domains
of desynchronization, see Figure 11, right plot.

An increase of the stimulus amplification parameter \(K\)
results in a gradual decay of the order parameter \(R\)
( Figure 12, upper plot). This enables
a precise control of the extent of desynchronization in the
stimulated ensemble.
Simultaneously, the amplitude of the stimulation signal
\(\vert S(t) \vert\) decays as well, indicating a demand-controlled
character of the NDF stimulation.
For a fixed delay \(\tau > 0\)
the order parameter and the amplitude of the stimulation signal
decay according to the following power law as \(\vert K \vert\)
increases:
\[\tag{10}
R \sim \vert K \vert ^{-1/2}, \quad
\vert S \vert \sim \vert K \vert^{-1/2},
\]

see Figure 12, upper plot. The desynchronization
transition for increasing \(K\) also manifests itself in a sequence of
frequency-splitting bifurcations ( Figure 12, bottom plot),
where the observed individual
frequencies \(\overline{\omega}_{j} = \langle \dot{\psi}_{j} \rangle\)
of the stimulated oscillators split, one after another away from the
mean frequency \(\Omega\) and approach the natural frequencies \(\omega_j\) of the oscillators (black diamonds in Figure 12, bottom plot) as \(K\) increases.

For large values of \(K\) all stimulated oscillators
thus rotate with the frequencies
close to the natural ones and exhibit a uniform desynchronous dynamics without any kind
of cluster states. Simultaneously, depending on the values of the delay
\(\tau\ ,\) the NDF can significantly change
the mean frequency \(\Omega\ ,\) i.e., the frequency of the mean field
\(Z(t)\) of the stimulated ensemble
(Popovych et al. 2005, Popovych et al. 2006a).
The macroscopic dynamics can thus be either accelerated or slowed down,
whereas the individual dynamics remains close to the original one
(without stimulation and coupling). This opens up an approach for a
frequency control of the oscillatory population stimulated with
the NDF.

The above properties, namely, the robustness with respect to parameter variation
(compare Figure 11 to Figure 9),
precise control of the extent of synchronization, and the frequency control
distinguish the NDF desynchronizing method from the linear single-site delayed feedback.
Another peculiarity of NDF is an indirect control of synchronization as discussed below.

In the case of two (or more) oscillator populations interacting
according to a drive-response coupling scheme, a particular stimulation
setup, called mixed nonlinear delayed feedback,
can be applied (Popovych & Tass 2010)
( Figure 13, upper plot). The coupling within
population \(2\)
is assumed to be weak, so that -- isolated from population \(1\) --
no synchronization emerges in population \(2\ .\)
In contrast, the coupling in population \(1\) is strong enough
to cause synchronization within population \(1\ .\) It then drives the
second population, which synchronizes because of the driving and
sends a response signal back to population \(1\ .\)
The second, driven ensemble is stimulated with signal \(S(t)\ ,\)
which is constructed from a linear combination
\(S_{\varepsilon}\) of the mean fields \(Z\)
and \(W\) of
populations \(1\) and \(2\ ,\) respectively, according to the
rule of the NDF, see Eq. (8)
\[\tag{11}
S(t) = K S_{\varepsilon}^2(t) S^{\ast}_{\varepsilon}(t - \tau), \quad
S_{\varepsilon}(t) = \varepsilon Z(t) + (1-\varepsilon)W(t).
\]

The level of mixing of the mean fields \(Z\) and \(W\) within the
stimulation signal is given by the parameter \(\varepsilon\ .\)
If \(\varepsilon = 0\ ,\) only the driven and stimulated
population \(2\)
is measured, and if \(\varepsilon = 1\ ,\) only the drive population
\(1\) contributes to the stimulation signal.
For intermediate values of the mixing \(\varepsilon \in (0,1)\)
the mixed dynamics of both ensembles is used as stimulation signal.
Such a mixing signal can be used for stimulation in the case where only
different linear combinations of the mean fields of the drive and
response ensembles can be measured, e.g., due to electrophysiological
conditions.

Proportional-integro-differential feedback

Figure 14: PID Control: The mean field is measured in one part of the controlled ensemble and, after processing according to the proportional-integro-differential algorithm, it is administered to the other part of the ensemble.

For a particularly difficult situation, where the measurement and
stimulation are not possible at the same time and at the same place,
there is another control method which is based on
a proportional-integro-differential (PID) feedback. The scheme of
this stimulation protocol is sketched in Figure 14.
The controlled ensemble of \(N\) coupled oscillators is split into two
separate sub-populations of \(N_{1}\)
and \(N_{2} = N - N_{1}\) oscillators,
one being exclusively measured and the other being exclusively stimulated.
In this way a separate stimulation-registration setup is realized,
where the recording and stimulating sites are spatially separated,
and the measured signal is not corrupted by stimulation artifacts.
The observed signal is considered to be the mean field \(Z_{1}\) of
the measured sub-population. Below, only the proportional-differential
(PD) feedback is illustrated (for more details, see (Pyragas et al. 2007)).
The stimulation signal \(S(t)\) which is delivered to the second, stimulated
sub-population is constructed as

\[\tag{12}
S(t) = P Z_{1}(t) + D\dot{Z_{1}}(t),
\]

where the parameters \(P\) and \(D\) define
the strength of the proportional and differential feedback,
respectively. In the framework of the phase oscillators
(1) one arrives to the following equation for
the phases \(\psi_{j}\ :\)

Figure 15: Upper plot: The time averaged order parameter \(\langle R \rangle\) of the whole population (13), (14) versus the strength of the PD feedback (with \(P = D\)) for different splitting ratios \(N_{1}:N_{2}\) and different mean frequencies \(\Omega\ .\) Bottom plot: The values of the order parameter \(\langle R \rangle\) (encoded in color) of the measured sub-population (left) and stimulated sub-population (right) versus stimulation parameters \(P\) and \(D\ .\) The white curve is the parameter threshold of the onset of desynchronization in both sub-populations obtained in (Pyragas et al. 2007).

where \(F_j\) is the corresponding phase representation of the above
stimulation signal \(S\) and \(H(\cdot)\) is the Heaviside function
defined as \(H(k)=0\) if \(k\leq0\)
and \(H(k)=1\) if \(k>0\ .\)

The effect of the stimulation with PD feedback is illustrated
in Figure 15. As the strength of the feedback
(parameters \(P\) and \(D\)) increases the stimulation results in a
complete desynchronization of both, measured and stimulated
sub-populations. The threshold of the onset of desynchronization
depends on the splitting ratio \(N_{1}:N_{2}\) of the size of the
sub-populations and on the mean frequency \(\Omega\ :\)
The threshold is larger for a smaller number of oscillators \(N_2\)
in the stimulated population or for larger frequency \(\Omega\ .\)
The later dependence can be eliminated if an integral component
is included in the stimulation signal, see (Pyragas et al. 2007).
Moreover, if the coupling in the ensemble is rather weak,
the desynchronization can be achieved by applying the
proportional feedback only. In contrast, in the case of
strong coupling the stimulation signal must also contain
the differential feedback for robust desynchronization.

Effects of desynchronizing stimulation in the presence of synaptic plasticity

Desynchronizing stimulation may decrease the strength of the neurons' synapses by decreasing the rate of coincidences.

Neuronal networks with STDP may exhibit bi- or multistability.

By decreasing the mean synaptic weight, desynchronizing stimulation
may shift a neuronal population from a
stable synchronized (pathological) state to a stable desynchronized
(healthy) state, where the neuronal population
remains thereafter.

Figure 16: Illustration of the effects of kindling and anti-kindling stimulation on a population of neuronal bursters: the local field potential \(LFP\) (top), the mean synaptic connectivity (middle) and the synchronization measure (bottom) are plotted as a function of time. Periods ON stimulation are indicated by red bars and vertical lines. Anti-kindling coordinated reset stimulation is performed in a demand-controlled manner here. The same results are, however, obtained with open-loop coordinated reset stimulation. For details of the model and its parameters see (Hauptmann & Tass 2007, Tass & Hauptmann 2007).

The kindling and anti-kindling processes are illustrated in Figure 16. A kindling stimulation,
i.e., spatially homogeneous periodic low-frequency stimulation,
is first administered to the initially
desynchronized population of bursting neurons. The synaptic
connectivities are modified following a simplified synaptic plasticity rule
with symmetric spike timing characteristics
(Debanne et al. 1998, Magee & Johnston 1997, Abbott & Nelson 2000,
Kepecs et al. 2002, Wittenberg & Wang 2006, Pfister & Gerstner 2006).
The stimulation induces an increase of the rate of coincident bursts,
which, in turn, results in an increase of the corresponding synaptic
connections, see Figure 16 (middle plot).
Induced by the kindling stimulation a synchronized state is established,
which stably persists if left unperturbed. The stimulation thus shifts the
population from a stable desynchronized state (modeling a healthy state)
characterized by rather weak coupling coefficients to a stable
synchronized states (modeling disease states) of strongly
coupled oscillators. In a second step, an anti-kindling stimulation,
i.e., desynchronizing coordinated reset stimulation, is applied
to the kindled population (Hauptmann & Tass 2007, Tass & Hauptmann 2007).
It results in a reduction of the rate of coinciding bursts and leads
to a reduction of the synaptic connections, which, finally,
ends up in a stabilization of a desynchronized state which
also persists thereafter if left unperturbed, see Figure 16.

Desynchronizing stimuli (Tass & Majtanik 2006, Hauptmann & Tass 2007,
Tass & Hauptmann 2007)
have the potential to shift the population from the basin of attraction
of a stable synchronized state into the basin of attraction of a stable
desynchronized state. This concept might have substantial impact on
novel therapeutic stimulation strategies for the therapy of
neurological and psychiatric diseases characterized by abnormal
synchrony. In fact, long-lasting desynchronizing effects of coordinated
reset stimulation have been observed in rat hippocampal slice rendered
epileptic by magnesium withdrawal (Tass et al. 2009). Another aspect
which is relevant from the neuroscientific and, especially,
clinical standpoint is the stimulation effect on physiological
synaptic connections. In particular, do physiological
(e.g., sensory input-related) synaptic connections survive
the desynchronization-induced anti-kindling, or are they erased
together with pathological (i.e., up-regulated) connections?
In a first computational study (Hauptmann & Tass 2010)
CR stimulation turned out to restore
physiological synaptic connections during the anti-kindling process.